The characteristic lifetime of \(1.10(2)\,\) ns is extracted from a Gaussian fit. The lifetime is limited by inhomogenous Doppler broadening of the atoms causing dephasing of the atomic polarization. Atom \(j\)  constituting the collective dipole moment between the ground and storage state has a net wavevector \(e^{i(\vec{k}_s - \vec{k}_c)\cdot \vec{v}_jt}\)  where \((\vec{k}_s - \vec{k}_c)\cdot \vec{v}_j\)  is the Doppler shift for the \(j^\textrm {th}\)  atom. Given the broad Maxwell-Boltzmann distribution of velocities of the atomic vapour, each atom accumulates a different phase thereby reducing the collective dipole moment. The spatial wavelength of the collective orbital-wave excitation is around \(1.6\,\mu \) m, a distance traversed by an atom in the hot vapour in a few nanoseconds. The storage time is also reduced by the interference of multiple pathways through different hyperfine states with characteristic timescales of \(\sim 10\) ns [1]}. The natural lifetime of the storage state provides the ultimate limit of \(\sim \,90\,\) ns [2]}, [3]}. The timescale we have measured here is consistent with that expected from Doppler dephasing [4]} suggesting strongly that this is the main limitation for our storage time.
